Derivative of b^x
b^x ln(b)
The derivative of b^x, where b is a constant greater than 0 and not equal to 1, is:
d/dx (b^x) = ln(b) * b^x
To understand why this is the case, we can use logarithmic differentiation. Let y = b^x. Taking the natural logarithm of both sides, we get:
ln(y) = ln(b^x) = x * ln(b)
Now we can implicitly differentiate with respect to x:
d/dx ln(y) = d/dx (x * ln(b))
1/y * dy/dx = 1 * ln(b)
dy/dx = y * ln(b)
Substituting y = b^x, we get:
dy/dx = b^x * ln(b) = ln(b) * b^x
Therefore, the derivative of b^x is ln(b) * b^x.
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